Questions:
I understand this proof and why b[tex]\in[/tex]B is fixed, but to prove [tex]|A \times C| \leq |(A \times B) \times C|,[/tex] it is not necessary to state to fix b. Since b can be any value in the triple order, it doesn't have to be fixed, in fact it can hold any value from the set B and still fulfill the condition: [tex]|A \times C| \leq |(A \times B) \times C|[/tex]. If what I am saying is true, is there an easy modification of the proof above? Personally I think by fixing b, we are limiting the scope of the proof.